The proof of the Cwikel estimate (REF ) for \(p=2\)  is elementary and is presented in the proof of [1]}. In fact, in this case we actually have an equality,
\(\Vert \lambda (x)g(-i\nabla )\Vert _{\mathcal {L}_2} = \Vert x\Vert _{L_2}\Vert g\Vert _{\ell _2}, \qquad x\in L_2(n_\theta ), \ g\in \ell _2(\mathbb {Z}^n).\) 
In particular, Hypothesis 3.1 of [2]} is satisfied with a constant equal to 1.
It follows from [2]} that if \(x \in L_2(n_\theta )\)  and \(g\in \ell _\infty (\mathbb {Z}^n)\) , then
\(\mu (\lambda (x)g(-i\nabla ))^2 \prec \prec 130\mu (x\otimes g)^2.\) 
Thus, if \(x\in L_\infty (n_\theta )\)  and \(g\in \ell _p(\mathbb {Z}^n)\)  are positive, then by using (REF ) we get
\(\big \Vert |\lambda (x)g(-i\nabla )|^2\big \Vert _{L_{\frac{p}{2}}} \le 130\big \Vert |x|^2\otimes |g|^2\big \Vert _{L_{\frac{p}{2}}\big (L_{\infty }(n_\theta )\overline{\otimes } \ell _{\infty }(\mathbb {Z}^n)\big )}\le 130 \Vert x\Vert _{L_p}^2\Vert g\Vert _{\ell _p}^2.\) 
As \( \Vert |\lambda (x)g(-i\nabla )|^2\Vert _{L_{\frac{p}{2}}} = \Vert \lambda (x)g(-i\nabla )\Vert _{L_{p}}^2\)  we get
\(\big \Vert \lambda (x)g(-i\nabla )\big \Vert _{L_p} \le \sqrt{130} \Vert x\Vert _{L_p}\Vert g\Vert _{\ell _p}.\) 
Let \(N\ge 1,\)  and let \(\theta ^{\oplus N}\)  be the \((Nn)\times (Nn)\) -matrix given by the direct sum of \(N\)  copies of \(\theta \) . Note that
\(\Vert x\Vert _{L_p(n_\theta )} = \Vert x^{\otimes N}\Vert _{L_p\big ({Nn}_{\theta ^{\oplus N}}\big )}^{\frac{1}{N}}, \qquad \Vert g\Vert _{\ell _p(\mathbb {Z}^n)} = \Vert g^{\oplus N}\Vert _{\ell _p(\mathbb {Z}^{Nn})}^{\frac{1}{N}}.\) 
Moreover, as \( L_2({Nn}_{\theta ^{\oplus N}}) = L_2(n_\theta )^{\otimes N}\) , where \(\otimes \)  is the Hilbert space tensor product, we also have
\(\Vert T\Vert _{{L}_p(n_\theta )} = \Vert T\Vert _{{L}_p\big (L_2\big ({Nn}_{\theta ^{\oplus N}}\big )\big )}^{\frac{1}{N}} \qquad \forall T\in {L}_p(L_2(n_\theta )).\) 
The inequality (REF ) holds on \(n_{\theta ^{\oplus N}}\) . Thus, if \(x\in L_p(n_\theta )\)  and \(g\in \ell _p(\mathbb {Z}^n)\) , then
\(\big \Vert \ (\lambda (x)g(-i\nabla ))^{\otimes N}\big \Vert _{{L}_{p}\big (L_2\big ({nN}_\theta \big )\big )}& =\big \Vert \lambda (x^{\otimes N})g^{\otimes N}(-i\nabla )\big \Vert _{\mathcal {L}_{p}(L_2({nN}_\theta ))}\\& \le \sqrt{130}\big \Vert x^{\otimes N}\big \Vert _{L_p\big ({Nn}_{\theta ^{\oplus N}}\big )}\big \Vert g^{\oplus N}\big \Vert _{\ell _p(\mathbb {Z}^{Nn})}.\) 

Combining this with (REF )–(REF ) gives
\(\big \Vert \lambda (x)g(-i\nabla )\big \Vert _{{L}_p} \le 130^{\frac{1}{2N}}\Vert x\Vert _{L_p}\Vert g\Vert _{\ell _{p}}.\) 
Letting \(N\rightarrow \infty \)  then yields
\(\big \Vert \lambda (x)g(-i\nabla )\big \Vert _{{L}_p} \le \Vert x\Vert _{L_p}\Vert g\Vert _{\ell _{p}}.\) 
This is the inequality (REF ).
To estimate the weak Schatten norms we can do a similar argument using (REF ) in place of (REF ). Let \(p>2\) . By using (REF ) and (REF ) and arguing as above shows that if \(x\in L_\infty (n_\theta )\)  and \(g\in \ell _{p,\infty }(\mathbb {Z}^n)\)  are positive, then
\(\big \Vert \lambda (x)g(-i\nabla )\big \Vert _{{L}_{p,\infty }} \le \left(\frac{130p}{p-2}\right)^{\frac{1}{2}}\Vert x\Vert _{L_p}\Vert g\Vert _{\ell _{p,\infty }}.\) 
As above the inequality continues to hold if \(x\)  and \(g\)  are not positive. Thanks to the density of \(L_\infty (n_\theta )\)  in \(L_p(n_\theta )\) , it follows that, if \(x\in L_p(n_\theta )\)  and \(g\in \ell _{p,\infty }(\mathbb {Z}^n)\) , then \(\lambda (x)g(-i\nabla )\in {L}_{p,\infty }\)  and the inequality (REF ) holds. This gives the 2nd part of Theorem REF . The proof is complete.
